In mathematics, the perimeter is the distance around a two-dimensional shape. The formula for finding the perimeter of certain shapes will be discussed in this lesson, and there will be some examples to help you understand how to calculate the perimeter.

Definition

The word perimeter means a path that surrounds an area. It comes from the Greek word 'peri,' meaning around, and 'metron,' which means measure. Its first recorded usage was during the 15th century. In mathematics, the perimeter refers to the total length of the sides or edges of a polygon, a two-dimensional figure with angles. When describing the measurement around a circle, we use the word circumference, which is simply the perimeter of a circle.

There are many practical applications for finding the perimeter of an object. Knowing how to find the perimeter is useful for finding the length of fence needed to surround a yard or garden, or the amount of decorative border to buy to cover the top edges of a room's walls. Also, knowing the perimeter, or circumference, of a wheel will let you know how far it will roll through one revolution.

Perimeter Formulas: Some Terms

The basic formula for finding the perimeter is just to add the lengths of all the sides together. However, there are some specialized formulas that can make it easier, depending on the shape of the figure. Before we begin, let's define some abbreviations, or variables, we'll be using in our formulas.

We'll represent the perimeter, the value we're trying to find, with a capital P. For a shape that has all of its sides the same length, we'll use an s to represent a side. We can also use s with a number after it to represent sides of shapes that have more or less than four sides, which may be the same or different lengths. We can write these variables like this: s1, s2, s3, etc.

For a shape that has two of its opposite sides the same as each other and its other two opposite sides the same as each other but different from the first two sides, we'll need two variables. We'll call the longer of the two distances 'length' and the shorter of the two distances 'width.' We'll represent length with an l and width with a w, as follows:

l = length

w = width

Finding the Perimeter for a Square

Each side of a square has the same length, so we can use our abbreviation s to represent a side. A square has four sides, so we can find its perimeter by finding the length of any side and multiplying it by 4. We write the formula this way: P = 4s.

In the picture shown here, each side of the square has a length of 6 feet. Using our formula, P = 4s, we plug in the value of the length of one side for s: P = 4 * 6 ft. 6 * 4 = 24, so the perimeter of our square is 24 feet.

Finding the Perimeter for a Rectangle

A rectangle has right angles like a square does, but it has two longer sides that are the same (length) and two shorter sides that are the same (width). If we know the length of one side and the width of another, we can add them together and multiply by 2. We write the formula this way: P = 2(l + w).

To find the perimeter of the rectangle shown here, we need to have the length of one of the longer sides and the width of one of the shorter sides. We see from the labels that the length is 6 and the width is 3.

Starting with our formula, P = 2(l + w), we then substitute 6 for the l and 3 for the w: P = 2(6 + 3). Adding 6 and 3 equals 9, so our equation now looks like this: P = 2(9). Multiplying 2 times 9 gives us 18, which is the perimeter of the rectangle.

Finding the Perimeter of a Triangle

A triangle has three sides, which may be the same or different lengths. The easiest way to find the perimeter is to just add the sides together. We can write the formula like this: P = s1 + s2 + s3. For a triangle, we often represent the three sides with the letters a, b, and c, so we can also write the formula as P = a + b + c.

For the triangle shown here, we start with our formula and then plug in the lengths of each side in place of the variables representing the sides. Now, we add up the lengths of the sides. Adding 4 + 8 + 11 = 23, so the perimeter of our triangle is 23 centimeters.

Finding the Perimeter of a Polygon

A triangle is a polygon with only three sides. To find the perimeter of a polygon, add up the lengths of all the sides, just as you did for the triangle. The difference here is that the polygon shown here has more sides than the triangle did.

This polygon, called a pentagon, is a polygon with five sides, so we write the formula this way: P = s1 + s2 + s3 + s4 + s5. Now, we substitute the lengths of the sides for the variables representing them: P = 5 + 4 + 2 + 7 + 1. We add up the sides: 5 + 4 + 2 + 7 + 1 = 19, so we write our result this way: P = 19.

One thing to remember when finding the perimeter of an object is that you can only add lengths that have the same unit. If one side of an object is in inches and another is in feet, you must convert either inches to feet or feet to inches before adding. If no units are given, you can assume they are the same. Now, we're going to work through a couple of example problems on perimeters.

Example Problem 1

Andrew is going to build a wooden hat box. He decides that each side should be 5 inches long. He also decides to make the box so that the lid and bottom are the shape of a regular hexagon. What will the perimeter of the lid be?

Before we answer this question, we must first define some terms:

A hexagon is a six-sided figure.

A regular hexagon is a hexagon where all six sides are the same length.

Each side is 5 inches long, so we replace each of the side-length variables with 5 inches, like so: P = 5 in + 5 in + 5 in + 5 in + 5 in + 5 in.

Then, we can find the perimeter by adding up the sides. The perimeter of the box is 5 + 5 + 5 + 5 + 5 + 5, or 30 inches. P = 30 in.

Example Problem 2

Paula wants to fence in an area to make a rectangular garden, but doesn't want to spend any money to buy fencing. She has 36 feet of fencing left over from another project, and she knows she wants her garden to be 10 feet long. What can the width of her garden be if she uses all the fencing she has?

Starting with the information that Paula wants a rectangular garden, we can use the formula for the perimeter of a rectangle: P = 2(l + w).

Then, we fill in what we know. The length of the garden is to be 10 feet, and its perimeter is to be 36 feet because that's how much fencing Paula has. Now, our formula looks like this: 36 = 2(10 + w).

To solve for w (which will be the width of her garden), we multiply both values inside the parentheses by 2. Now our equation looks like this: 36 = 20 + 2w.

Subtract 20 from both sides: 36 - 20 = 20 + 2w - 20. 16 = 2w.

Finally, divide both sides by 2. 8 = w, or putting the variable on the left, we write: w = 8. Paula's garden will be 8 feet wide.

Lesson Summary

The perimeter is the distance around the outer edge of a two-dimensional figure. To find the perimeter, you need to know the length of one or more sides, depending on the shape of the figure. Finding the perimeter has many practical applications, including finding how much material is needed for a building project.

Vocabulary Terms & Definitions

Perimeter

Terms

Definitions

Perimeter

the path that surrounds an area

Polygon

two-dimensional figure with angles

Circumference

the perimeter of a circle

Pentagon

a polygon with five sides

Hexagon

a six-sided figure

Regular hexagon

a hexagon in which all six sides are the same length

Learning Outcomes

Complete this video lesson and transcript on perimeter with these goals in mind:

Additional Perimeter Examples

In the following examples, students will gain practice calculating the perimeter of various shapes and use the definition of perimeter to solve for missing side lengths. After completing the examples, students will understand that the perimeter of a shape is the distance around the shape - regardless of if the shape is a "standard" geometric shape or a really complicated shape.

Examples

1. Find the perimeter of a triangle, in inches, with side lengths: 5 inches, 1 foot, and 13 inches.

2. What side length of a square would be needed so that the perimeter of the square equals the perimeter of an equilateral triangle with side length 9?

3. The length of a rectangle is 3x + 2 and the width of the rectangle is 2x - 1. If the perimeter is equal to 52, find the length and width of the rectangle.

4. If the perimeter of the shape below is equal to 73 inches, what is the total length of the curved portion of the shape?

Solutions

1. To find the perimeter of the triangle, simply add up the side lengths. Be careful - all side lengths must have the same units, so we will convert the 1-foot side length to 12 inches. Then, the perimeter is P = 5 + 12 + 13 = 30 inches.

2. First find the perimeter of the equilateral triangle: P = 9 + 9 + 9 = 27. We want to find the side length of a square with perimeter 27. The formula for the perimeter of a square is P = 4s so we have 27 = 4s and so s = 27/4 = 6.75.

3. Using P = 2(l + w), we have 52 = 2(3x + 2 + 2x - 1). Dividing by 2 and then combining like terms, we have 26 = 5x + 1. So, 25 = 5x and so x = 5. Then, substitute x = 5 back into the formulas for length and width to find that the length is 3(5) + 2 = 17 and the width is 2(5) - 1 = 9.

4. The perimeter is found by adding up all the sides of the shape. Let x stand for the length of the curved portion of the shape. Then, we have 10.1 + 4 + 5 + 13 + 7.3 + 11.2 + x = 73. Adding the sides together, we have 50.6 + x = 73 and so the length of the curved portion is x = 22.4.

Summary:

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